In these lessons, we will learn• the rules of the Locus theorem • exactly how the rules of the Locus Theorem have the right to be used in real world examples.• exactly how to recognize the locus of clues that will certainly satisfy an ext than one condition.

You are watching: The locus of all points that are the same distance from two given points is a

Related PagesLoci In Geometry Loci an ext Geometry class

The adhering to diagrams give the locus of a allude that meet some conditions. Role downthe page for an ext examples and solutions.


When a suggest moves in a airplane according to part given conditions the route along which itmoves is dubbed a locus. (Plural the locus isloci.).


A allude P moves such that it is constantly m devices from the suggest Q.

Locus formed: A circle with center Q and also radius m.


Example:Construct the locus the a allude P at a continuous distance that 2 centimeter from a fixed suggest Q.

Solution:Construct a one with center Q and radius 2 cm.



A point P move such that it is equidistant kind two fixed pointsX and Y.

Locus formed: A perpendicular bisector that the heat XY.


Example:Construct the locus of point P relocating equidistant from resolved points X and also Y and XY = 6 cm.

Solution:Construct a perpendicular bisector the the line XY.



A allude P moves so the it is constantly m devices from a directly line AB.

Locus formed: A pair the parallel lines m unitsfrom AB.


Example:Construct the locus that a point P that moves a consistent distant of 2 centimeter from a right line AB.

Solution:Construct a pair of parallel present 2 centimeter from AB.



A point P moves so the it is constantly equidistant from twointersecting lines abdominal and CD.

Locus formed: angle bisectors of angles betweenlines abdominal and CD.


Example:The following number shows two right lines abdominal muscle and CD intersecting at point O. Constructthe locus of allude P such that it is constantly equidistant from ab and CD.


Example:Construct angles bisectors the angles in between lines abdominal muscle and CD.


Five an essential Locus Theorems and How To use Them

Locus theorem 1: The locus of points in ~ a fixed distance, d, from the point, p is a circlewith the given point P as its center and d as its radius.Locus organize 2: The locus that the points at a addressed distance, d, from a line, l, is a pairof parallel present d street from l and on either side of l.Locus theorem 3: The locus of clues equidistant from 2 points, P and also Q, is theperpendicular bisector of the heat segment figured out by the 2 points.Locus theorem 4: The locus of point out equidistant from two parallel lines, l1and l2, is a line parallel to both l1 and also l2 and midwaybetween them.Locus organize 5: The locus of clues equidistant from 2 intersecting lines, l1and l2, is a pair of bisectors that bisect the angles developed by l1and l2.

Example 1:A endowment map reflects a treasure covert in a park near a tree and also a statue. Themap indicates that the tree and also the stature space 10 feet apart. The treasure is buried 7 feetfrom the basic of the tree and additionally 5 feet native the base of the stature. How numerous places arepossible areas for the treasure to it is in buried? draw a diagram of the endowment map, andindicate v an X each possible location that the treasure.

Example 2:The distance between the parallel heat l and m is 12 units. Suggest A is on heat l.How countless points room equidistant from lines l and also m and also 8 systems from suggest A.

Example 3:Maria’s backyard has two trees that room 40 feet apart. She desires to placelampposts so the the the write-ups are 30 feet native both of the trees. Attract a map out to showwhere the lampposts might be put in relationship to the trees. How numerous locations for thelampposts room possible?

Five rules Of Locus Theorem utilizing Real world Examples

Locus is a set of points that accomplish a provided condition.There room five an essential locus rules.Rule 1: offered a point, the locus of point out is a circle.Rule 2: offered two points, the locus of points is a straight line midway between the two points.Rule 3: given a right line, the locus of point out is 2 parallel lines.Rule 4: provided two parallel lines, the locus of clues is a line midway in between the twoparallel lines.Rule 5: offered two intersecting lines, the locus of clues is a pair of present that reduced theintersecting present in half.

Intersection Of two Loci

Sometimes you might be forced to determine the locus of a suggest that satisfies two or moreconditions. We can do this by constructing the locus because that each the the conditions and also thendetermine wherein the 2 loci intersect.

Example:Given the line abdominal and the suggest Q, uncover one or more points that space 3 cm from abdominal muscle and 5 cmfrom Q.


Solution:Construct a pair the parallel lines 3 centimeter from line AB. Draw a circle with facility Q andradius 5 cm.


The clues of intersections are indicated by clues X and also Y.

It way that the locus is composed of the 2 points X and Y.

Example:Given a square PQRS through sides 3 cm. Build the locus the a allude which is 2 cm from Pand equidistant from PQ and also PS. Note the points as A and B.

Solution:Construct a one with facility P and radius 2 cm. Since PQRS is a square the diagonal PRwould it is in the angle bisector the the angle formed by the currently PQ and also PS. The diagonal whenextended intersects the circle at points A and also B.


Note: A common mistake is to identify only onepoint as soon as there can be another point which might be uncovered by prolonging the constructionlines or arcs; as in the over examples.

GCSE Maths Exam inquiries - Loci, Locus and also Intersecting Loci


Draw(i) the locus the a suggest that moves so the it is always exactly 4 cm from the resolved pointX and(ii) the locus the points much less than 4 centimeter from the fixed suggest X.

Draw the locus of clues no more than 3 centimeter from A and also no additional than 4 cm from B.

Draw the locus of a point exactly 3 centimeter away from directly line AB.

Draw (i) the locus of a allude equidistant native the points X and also Y.(ii) the locus of clues closer to the suggest X 보다 the suggest Y.(iii) the locus of clues closer to X 보다 Y however no less than 5 centimeter from X.

Draw the locus of point out closer to the line abdominal than the line BC in the rectangle ABCD.

See more: Whats Half Of 1/8 + 1/2? What Is A Half Of 1 4 In Fractions

A dog is ~ above a command tethered to a write-up in the edge of a garden. The lead is 5 m long.A cat is complimentary to roam all components of the garden but is not enabled within 3 m of the houseby that owner. Present the for sure area the the cat deserve to safely roam top top the chart below.

Try the free Mathway calculator and problem solver below to practice assorted math topics. Shot the provided examples, or kind in your very own problem and check your answer with the step-by-step explanations.